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A new framework is presented for both understanding and developing graph-cut-based combinatorial algorithms suitable for the approximate optimization of a very wide class of Markov random fields (MRFs) that are frequently encountered in computer vision. The proposed framework utilizes tools from the duality theory of linear programming in order to provide an alternative and more general view of state-of-the-art techniques like the alpha-expansion algorithm, which is included merely as a special case. Moreover, contrary to alpha-expansion, the derived algorithms generate solutions with guaranteed optimality properties for a much wider class of problems, for example, even for MRFs with nonmetric potentials. In addition, they are capable of providing per-instance suboptimality bounds in all occasions, including discrete MRFs with an arbitrary potential function. These bounds prove to be very tight in practice (that is, very close to 1), which means that the resulting solutions are almost optimal. Our algorithms' effectiveness is demonstrated by presenting experimental results on a variety of low-level vision tasks, such as stereo matching, image restoration, image completion, and optical flow estimation, as well as on synthetic problems.